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What I want to do in this
video is talk a little bit about polynomial end behavior. And this is really
just talking about what happens to a polynomial if
as x becomes really large or really, really,
really negative. For example, we're familiar
with quadratic polynomials where y is equal to ax
squared plus bx plus c. We know that if a
is greater than 0, this is going to be an upward
opening parabola of some kind. So it's going to look
something like that, the graph of this equation,
or of this function, you could say. And if a is less
than 0, it's going to be a downward
opening parabola. We've spent less time with
third degree polynomials, but we've also seen
those a little bit. So for example, if you have
the third degree polynomial, y is equal to ax to the third
plus bx squared plus cx plus d, if a is greater than 0-- I don't
want to use that brown color. If a is greater than 0,
when x is really, really, really negative, this whole
thing is going to be really, really, really negative. And then it's going to increase
as x becomes less negative. It's going do
something-- it might do a little bit of
funky stuff in between. But then as x becomes more
and more and more positive, it will become more and more
and more positive as well. So it might look something like
this when a is greater than 0. But what about when
a is less than 0? Well then, just like
here, we would flip it. We would flip it so that
if a is less than 0, when x is really
negative, you're going to multiply that
times a negative a and you're going to
get a positive value. So it's going to look
something like this. And then it's going
to go like this. It might do a little bit of this
type of business in between. But then its end behavior,
it starts decreasing again. It starts decreasing. So when we talk
about end behavior, we're talking about the idea
of what is this function? What does this polynomial do
as x becomes really, really, really, really positive
and as x becomes really, really, really, really negative? And kind of fully recognizing
that some weird things might be happening in the middle. But we just want
to think about what happens at extreme values of x. Now obviously for
the second degree polynomial nothing really
weird happens in the middle. But for a third
degree polynomial, we sort of see that
some interesting things can start happening
in the middle. But the end behavior for
third degree polynomial is that if a is
greater than 0-- we're starting really small,
really low values-- and as a becomes positive,
we get to really high values. If a is less than 0
we have the opposite. And these are kind of the two
prototypes for polynomials. Because from there we can
start thinking about any degree polynomial. So let's just think about the
situation of a fourth degree polynomial. So let's say y is equal
to ax to the fourth power plus bx to the third plus cx
squared plus dx plus-- I don't want to write e because e has
other meanings in mathematics. I'll say plus-- I'm really
running out of letters here. I'll just use f, although
this isn't the function f. This is just a constant
f right over here. So let's just think about
what this might look like. Let's think about
its end behavior, and we could think about it
relative to a second degree polynomial. So its end behavior, if x is
really, really, really, really negative, x to the fourth is
still going to be positive. And if a is greater than 0
when x is really, really, really negative, we're
going to have really, really positive values, just
like a second degree. And when x is really
positive, same thing. x to the fourth is
going to be positive, times a is still
going to be positive. So its end behavior
is going to look very similar to a second
degree polynomial. Now, it might do--
in fact it probably will do some funky
stuff in between. It might do something that looks
kind of like that in between. But we care about
the end behavior. I guess you could call
the stuff that I've dotted lined in the middle, this is
called the non-end behavior, the middle behavior. This will obviously be
different than a second degree polynomial. But what happens at the
ends will be the same. And the reason why, when
you square something, or you raise something
to the fourth power, you raise anything to any even
power for a very large-- as long as a is greater than 0,
for very large positive values, you're going to get
positive values. And for very large
negative values, you're going to get very
large positive values. You take a negative number,
raise it to the fourth power, or the second power, you're
going to get a positive value. Likewise, if a is
less than 0, you're going to have very similar
end behavior to this case. For a polynomial where
the highest degree term is even-- so this is
a is less than 0-- your end behavior when a is really,
really, really, really negative, this thing
is going to be really, really, really positive. We're going to be multiplying
it times a negative, so it's going to
be really, really, really, really negative. So it'll look like this. And likewise, when x is really,
really, really positive, you get the same thing. Because you're going to be
multiplying a positive times a, which is negative,
and in between it might be doing
something like that. But its end
behavior, you see, is very similar to a second
degree polynomial. So if you ignore this, its
end behavior is very similar. Now the same is true
for a fifth degree if you were to compare
it to a third degree. And the overall idea here is
what happens to this value when we get really large x's
or really small x's? Are we taking it
to an even power? In which case for either really
negative values or really positive values we're going
to get positive values. And then it depends what
our coefficient a is. Or are we taking
it to an odd power? So the general
idea-- and actually let me just do a fifth degree,
just to make the clear. So if I had something
of the form y is equal to ax to the fifth
plus bx to the fourth plus-- and it just went all the way--
I won't even have to write it. This thing, if a
is greater than 0, would look something like this. Its end behavior is very similar
to a third degree polynomial where a is greater than 0. At the end it would do this. Now, it might do kind of
some craziness like this. I have to get this right. So 1, 2, 3-- it might do some
craziness like this in between. But then for really large x's
it will look the same as ax to the third when a
is greater than 0. So once again, very,
very similar end behavior when a is greater than
0, and very similar end behavior when
a is less than 0. It would look like this. At the ends at a
negative value it will be positive
because this part is going to be really negative. But then it's going to be
multiplied by a negative to get a positive. And for really positive values
of x, it will be negative. Because once again, this a
term is going to be negative. And then what it does
in between-- at least for the sake of
this video-- we're not really thinking about. So the big takeaway
here-- and this is kind of a little
bit of a drum roll here, we're talking
about end behavior-- if you're looking at an
even degree polynomial it's going to have end
behavior like a second degree polynomial. If you ignore what happens
in the middle, what happens at really negative
values of x and really positive values of x is going to be
very similar to a second degree polynomial. And if your degree
is odd, you're going to have very similar
end behavior to a third degree polynomial. You might do all sorts of
craziness in the middle, but given for a given a,
whether it's greater than 0 or less than 0, you will
have end behavior like this, or end behavior like that.